86 CHAPTER 3. DETERMINANTS

First find the determinant of this matrix. This is seen to be 12. The cofactor matrix ofA is  −2 −2 6

4 −2 0

2 8 −6

 .

Each entry of A was replaced by its cofactor. Therefore, from the above theorem, the inverseof A should equal

1

12

 −2 −2 6

4 −2 0

2 8 −6

T

=

 − 16

13

16

− 16 − 1

623

12 0 − 1

2

 .

This way of finding inverses is especially useful in the case where it is desired to find theinverse of a matrix whose entries are functions.

Example 3.1.13 Suppose

A (t) =

 et 0 0

0 cos t sin t

0 − sin t cos t

Find A (t)

−1.

First note det (A (t)) = et. A routine computation using the above theorem shows thatthis inverse is

1

et

 1 0 0

0 et cos t et sin t

0 −et sin t et cos t

T

=

 e−t 0 0

0 cos t − sin t

0 sin t cos t

 .

This formula for the inverse also implies a famous procedure known as Cramer’s rule.Cramer’s rule gives a formula for the solutions, x, to a system of equations, Ax = y.

In case you are solving a system of equations, Ax = y for x, it follows that if A−1 exists,

x =(A−1A

)x = A−1 (Ax) = A−1y

thus solving the system. Now in the case that A−1 exists, there is a formula for A−1 givenabove. Using this formula,

xi =

n∑j=1

a−1ij yj =

n∑j=1

1

det (A)cof (A)ji yj .

By the formula for the expansion of a determinant along a column,

xi =1

det (A)det

∗ · · · y1 · · · ∗...

......

∗ · · · yn · · · ∗

 ,

where here the ith column of A is replaced with the column vector, (y1 · · · ·, yn)T , and thedeterminant of this modified matrix is taken and divided by det (A). This formula is knownas Cramer’s rule.